 Solution

The direct image sheaf is defined on an open set $V\subseteq Y$ byRestrictions are those of $\mathcal{F}$, and the sheaf gluing axiom follows by taking inverse images of an open cover. The morphism of ringed spaces supplies ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$. Thus $a\in {\mathcal{O}}_Y(V)$ acts on $s\in \mathcal{F}({\varphi }^{-1}V)$ by ${\varphi }^{\#}(a)s$, making ${\varphi }_{\ast }\mathcal{F}$ a sheaf of modules over ${\mathcal{O}}_Y$. This definition uses no quasi-coherence.